WO2023222208A1

WO2023222208A1 - Reducing kinematic error

Info

Publication number: WO2023222208A1
Application number: PCT/EP2022/063355
Authority: WO
Inventors: Arne WAHRBURG; Richard Roberts; Mikael NORRLÖF; Stig Moberg
Original assignee: Abb Schweiz Ag
Priority date: 2022-05-17
Filing date: 2022-05-17
Publication date: 2023-11-23

Abstract

A method is proposed for reducing kinematic error in a joint (5j) between a distal portion and a proximal portion of a manipulator (1), the joint (5j) having associated to it a motor (9j) mounted in one of said portions and coupled to the other one of said portions for driving rotation of the distal portion relative to the proximal portion by a reduction gear (10j). The method comprising the steps of: a. providing a kinematic correction generator (13j) which is programmable using a parameter vector (0, p'p") and outputting, based on said parameter vector and a position (q mot,j ) input, a kinematic correction (8qke, 8qke) which is a sum of at least one sinusoid having a frequency (k i) which is a frequency of rotation of the motor (9j) or of a periodic event in the reduction gear (10j) occurring at a higher frequency than the rotation of the distal portion, or an integer multiple thereof, and an amplitude and a phase defined by said parameter vector (0, p', p"), b. outputting to the motor a first drive control signal specifying a motor speed which is a sum of a standard speed (qmot,j) and a first kinematic correction (Eqke) output by the kinematic correction generator (13j) based on a first parameter vector (0); c. extracting, from an evaluation signal (T) received from the joint (5j) in response to the first drive control signal, at least one frequency component having the frequency of the at least one sinusoid, and determining a first feedback vector (p) defining a first feedback amplitude (Ai,j) and a first feedback phase (i9ii7) of said frequency component, d. outputting to the motor (9j) a second drive control signal specifying a motor speed (qmot,j + Eqke) which is a sum of the standard speed (qmot,j) and a second kinematic correction (Eqfke) output by the kinematic correction generator (13j) based on a second parameter vector (p'); e. extracting, from an evaluation signal (T) received from the joint (5j) in response to the second drive control signal, a frequency component having the frequency of the at least one sinusoid, and determining a second feedback vector (p) defining a second feedback amplitude (Ai,j) and a second feedback phase (Oi,j) of said frequency component, f. subtracting said first feedback vector (p) from said second feedback vector (p) to obtain a feedback difference vector (p); g. finding a transformation which transforms said feedback difference vector (p) into a difference (p'- 0) between second and first parameter vectors, h. applying the transformation to the second feedback vector (p) to obtain a third parameter vector (p"), programming the kinematic correction generator (13j) using the third parameter vector (p").

Description

Reducing kinematic error

The present invention relates to methods and apparatus for reducing kine- matic error when controlling the movement of a joint in a robot manipulator.

A robot manipulator comprises a plurality of links, which are coupled to each other, to a base or to an end effector by rotatable joints. Rotation of such a joint is driven by a motor, typically via a reduction gear. Play and in- accuracy in the manufacture of the motors and gears affects the accuracy with which the position and the speed of an end effector at the distal end of the robot arm can be controlled. Harmonic drive gears are widely used as reduction gears in robots, since they are practically play-free, but inevitable inaccuracies, such as eccentricity of a gearwheel or spline, would still cause the rotation speed of a joint to fluctuate even if its motor could be driven at a perfectly constant speed, so that the position of the joint can de- viate from what one would expect based knowing the position of the motor and the transmission ratio of the reduction gear. This deviation is commonly referred to as kinematic error or transmission error.

The object of the present invention is to provide simple and cost-efficient way of reducing kinematic error in a robot joint.

According to an aspect of the present invention, this object is achieved by a method for reducing kinematic error in a j-th joint between distal and proxi- mal portions of a manipulator, the joint having associated to it a motor mounted in one of said portions and coupled to the other one of said por- tions for driving rotation of the distal portion relative to the proximal portion by a reduction gear, the method comprising the steps of a. providing a kinematic correction generator which is program- mable using a parameter vector and outputting, based on said parameter vector and a position input, a kinematic cor- rection which is a sum of at least one sinusoid having a frequency which is a frequency of rotation of the motor or of a periodic event in the reduction gear occurring at a higher frequency than the rotation of the distal portion, or an integer multiple thereof, and an amplitude and a phase defined by said parameter vector, b. outputting to the motor a first drive control signal specifying a motor speed which is a sum of a standard speed and a first kinematic correction output by the kinematic correction gen- erator based on the first parameter vector; c. extracting, from an evaluation signal received from the joint in response to the first drive control signal, at least one fre- quency component having the frequency of the at least one sinusoid, and determining a first feedback vector defining a first feedback amplitude and a first feedback phase of said frequency component, d. outputting to the motor a second drive control signal specify- ing a motor speed which is a sum of the standard speed and a second kinematic correction output by the kinematic cor- rection generator based on the second parameter vector; e. extracting, from an evaluation signal received from the joint in response to the second drive control signal, a frequency component having the frequency of the at least one sinus- oid, and determining a second feedback vector defining a second feedback amplitude and a second feedback phase of said frequency component, f. subtracting said first feedback vector from said second feed- back vector to obtain a feedback difference vector; g. finding

transformation which transforms said feed- back difference vector into a difference between second and first parameter vectors, h. applying the transformation to the first feedback vector

to obtain a third parameter vector, i. programming the kinematic correction generator using the third parameter vector.

In a preferred embodiment, the above processing is simplified by choosing the first parameter vector to define an amplitude of the at least one sinusoid to be zero. The sinusoid can be defined as a weighted sum of sine and co- sine functions; in that case weighting coefficients of both would have to be zero. Alternatively it can be defined as a sine or cosine function having an amplitude and a phase; when the amplitude is zero here, the phase has no physical significance and can be assigned any value.

In order to facilitate processing of the evaluation signal, the standard speed should be constant or, at least, substantially constant.

The standard speed can be regarded as substantially constant if effects of external forces acting on the distal portion on the evaluation signal, due e.g. to the weight of the distal portion driving or opposing the movement of the distal portion, can be suppressed by high-pass filtering the evaluation sig- nal. Due to the reduction gear, the speed of rotation of the motor will be much higher than that of the joint. Since the external forces are usually pro- portional to a sinusoid of the joint angle, these can be suppressed by set- ting a cutoff frequency lower than the motor frequency and higher than the motor frequency multiplied with the reduction ratio of the reduction gear. The standard speed or a harmonic thereof should correspond to a reso- nance frequency of the manipulator, i.e. the motor frequency needed to drive the distal portion at the standard speed or a harmonic thereof, prefer- ably the second harmonic, should equal the resonance frequency.

This speed doesn’t have no be known in advance; rather, finding it can be a preparatory step of the method of the invention. Specifically, in a prepara- tory stage of the method, the distal portion can be controlled to rotate at various speeds, and the speed where vibration of the manipulator is strong- est can be chosen as the standard speed.

It can be assumed that a robot controller is trying to find the critical speed when a same movement of the distal portion is carried out repeatedly at dif- ferent speeds, the speed changing from one iteration to the next, and the speed difference between successive iterations occasionally changing its sign, and/or the amount of the speed difference decreasing on each change of sign.

Since the moment of inertia of the manipulator or of parts of it is variable depending on configuration, and for the method of the invention it is desira- ble to have a low critical speed, the distal portion can be controlled to as- sume a configuration in which its moment of inertia is maximized, and which should be maintained while the method is carried out.

The evaluation signal can represent any measurable quantity that is sus- ceptible to fluctuate due to an imperfection of the motor or the reduction gear. It could be the speed of rotation of the distal member itself, as meas- ured e.g. by a dedicated angle sensor built into the manipulator for measur- ing rotation of the joint on a downstream side of the reduction gear, a laser scanner or other optical sensing means for observing the manipulator from outside, or the like. According to a preferred embodiment, the evaluation signal is representative of motor torque, since suitable sensing circuitry ex- ists in many legacy manipulators, so that when the method of the invention makes use of this circuitry, it can be implemented in such manipulators by a control software update.

When a harmonic drive gear is used as a reduction gear, its wave genera- tor will usually be connected to the motor output shaft, whereas its flexspline is connected to the output shaft of the reduction gear, and the cir- cular spline is fixed. When the number of teeth of the circular spline is i_c , and that of the flexspline is i_f, the orientation of the wave generator relative to the flexspline becomes identical after every rotations of the wave

generator ant the motor output shaft, causing contributions to the kinematic error at times the rotation frequency of the motor, where m is a small

integer. Since is if is slightly smaller than i_c, contributions of the flexspline and the circular spline to the kinematic error tend to have frequencies that differ slightly from those of the motor and the wave generator.

Experiments have shown that for most harmonic drive gears the most im- portant contribution of the flexspline and the circular spline is at is 2

times the rotation frequency.

In order to take account of various sources contributing to the kinematic er- ror, in most practical cases the kinematic correction will be a sum of two or more sinusoids.

The kinematic correction generator can best be provided by programming a processor to operate as such, in particular by updating a control processor of a legacy manipulator to enable it to operate as the kinematic correction generator, or by updating the third parameter vector whenever mainte- nance or repair work has been carried out at the corresponding joint. According to another aspect of the present invention, the object is achieved by a robotic system comprising a manipulator having a proximal portion, a distal portion, a joint con- necting said proximal and distal portions and a motor for driving rotation of the joint, and a controller connected to the motor and adapted to carry out at least steps b. to i. of the method described above.

According to a further aspect, the object is achieved by a computer soft- ware product comprising a plurality of instructions which, when executed by a processor, cause the processor to carry out at least steps b. to i. of the method described above.

Further features and advantages of the invention will become apparent from the subsequent description of embodiments, referring to the appended drawings.

Fig.1 is a block diagram of a robotic system implementing the inven- tion;

Fig. 2 is a perspective view of the manipulator of the system of Fig. 1 ;

Fig. 3 is a cross section of a harmonic drive gear;

Fig. 4 is a pose assumed by the manipulator while carrying out the method for one of its joints; and

Fig. 5 is a pose assumed by the manipulator while carrying out the method for another of its joints. Fig. 1 is a block diagram of a robot system comprising an articulated robot arm 1 and a controller 2, in which the method of the invention can be ap- plied. The robot arm 1 , shown in more detail in Fig. 2, has a stationary base 3 fixed to a support, and a plurality of links 4j, j=1 , 6 rotatably connected to each other and to the base 3 by joints 5_j. The joint 5₅ connecting links 4₄, 4₅ is concealed behind link 4₅. Each joint 5_j enables a rotation of the link 4j on its distal side by an axis 6_j. Axis 6₁ is vertical, axes 6₂ and 6₃ are hori- zontal; the orientations of the other axes vary. Each joint 5_j has a motor as- sociated to it for driving its rotation; the motor can be provided in a member on the proximal side of the joint and be connected to a member on the dis- tal side thereof, or vice versa. In the robot arm of Fig. 2, joints 5₁ , 6₂, 6₃ have their motors in members on their proximal side, i.e. in base 3 and links 4₁ , 4₅, respectively; joints 5₃, 5₄ and 5₅ have the motors on their distal side, in links 4₃, 4₄ and 4₅, respectively.

Motors 9₂, 9₃, 9₄ associated to joints 5₂, 5₃, 64 and reduction gears 10₂, 10₃, 10₄ connecting these motors to links 4₂, 4₂ and 4₃, respectively, are shown in Fig. 1 . Each motor 9_j is provided with a current sensor 7_j for measuring its winding current. Since measurement data from sensor 7_j are representa- tive of a torque applied by the motor 9_j they will be referred to below as torque data

. A resolver 8_j is placed at each joint 5_j so as to measure its rotation angle

The reduction gear 10_j is a harmonic gear, shown in cross section in Fig. 3, in which a wave generator 16 is rigidly coupled to an output shaft 17 of motor 9_j, and is pressing teeth of a flexspline 18 into local engagement with a cir- cular spline 19 surrounding it. When the number of teeth of the circular spline 19 is i_c and that of the flexspline 18 is i_f, the transmission ratio of the gear Since the flexspline is 18 rotating in a direction opposite to

e generator 16, points of the two currently in contact will meet again after the wave generator 16 has rotated slightly less than

a complete turn.

Controller 2 comprises a trajectory generator 11 for outputting, with an ap- propriate timing, position and speed commands to joint controllers 12_j asso- ciated to a specific joint 5_j as would be required for having a reference point of link 4₆ or of the tool mounted to it follow a predetermined trajectory as- suming that the joints are free from kinematic error. Each joint controller 12_j further has associated to it a kinematic error calculator 13_j for calculating a kinematic error correction

and adders 14_j, 15_j for superimposing the kinematic error correction onto the commands

output by the trajectory generator 11 , and providing corrected commands

will be explained in detail below.

The method of the invention can be carried out for each joint 5_j at a time.

In a first step of the method of the invention (S1 , see Fig. 5), the manipula- tor 1 is brought into a configuration where a resonance frequency of it com- ponents on the distal side of joint 5_j is maximized by moving the centres of gravity of links 4_j-4₆ as far away from axis 6_j as possible, as shown in Fig. 4 for j=2. A counting index i is set to 1 , a fluctuation reference Fo is set to 0, and a step width

is initialized.

A rotation speed for motor 9_j is set (S2), and trajectory generator 11

controls a rotation around axis 6j at nominally constant speed by outputting angle commands q to motor 9_j that have a constant rate of change

In step 3, while the joint 5_j is being rotated at the speed set in step S2, im- perfections of the motor 9_j and the reduction gear 10j cause the current de- tected by sensor 7_j to fluctuate. A measure F_i of this fluctuation, e.g a standard deviation of currents measured while carrying out the rotation, is recorded (S4).

If in step S5, F_i-F_{i - 1}is found to be positive, and is above a pre-de-

termined threshold, i is incremented and

is incremented by

in step S6, and the method returns to step S3. If Fj-Fj.i is found to be negative, Aq is multiplied by a factor ε between 0 and -1 (S7) before proceeding to step S6. When successive multiplications by ε have caused

to drop below the threshold, the amount F of fluctuation has reached a maximum, and it can be assumed that is the critical speed of joint 5_j.

Each time step S3 is executed, a series of motor positions and associated torque data frorn sensor 7_j are recorded. The last one of these series,

comprising N pairs of measurement values

and motor or arm positions at which these were obtained is retained for evaluation. The positions may be taken from commands

of the tra- jectory generator, preferably they are derived from readings

from re- solver 8j taken simultaneously with the torque measurement values.

Optionally, a spectral analysis of this series may be carried out in order to find out the frequency components that provide the largest contribution to the fluctuation of the torque data. Else it may simply be assumed that com- ponents which have been found to provide an important contribution in other manipulators will do so here, too.

of M harmonics is fitted to the N collected data points with solving for the parame-

ter vector p with

wherein k_b i =

refers to the important frequency components men- tioned above. If it is known that the most important frequency components are e.g. at twice the rotation frequency of the motor 9_j and at times

the rotation frequency of the motor 9_j, we may set M = 2, k₂ = 2 for the second harmonic of the motor rotation and for a nearby contri-

bution of the harmonic gear. The parameter vector p, referred to below as the feedback parameter vector, contains a sine and cosine value

A_S,i,A_c,i for each frequency component (which is equivalent to characteriz- ing the component by an amplitude and a phase) and a constant offset C to handle a constant bias.

If the joint is subject to position-dependent gravity load (as is the case for joint 5₂ in Fig. 4), the torque data can be subjected to high-pass filtering (S8) prior to their evaluation in step S9. The transmission ratio of gear 10j being Rj, the filtering threshold frequency should be between the rotation frequency of motor 9_j and R_j times this frequency. In step S10, a second parameter vector arbitrarily defined,

and is transmitted to kinematic error calculator 13_j, thereby enabling it to output a kinematic error compensation

motor angle output by trajectory generator 11. In step S11 , another rotation around axis 6j is carried out, in which the motor receives commands to which compensations

have been

added in adders 14_j, 15_j. Torque data resulting

therefrom are collected, and are evaluated (S12) as described above in steps S8, S9 for torque data b, whereby a second feedback parameter vec- tor is obtained.

Processing of these data (S12) is described referring to Fig. 6. Fig. 6 is a series of diagrams showing components A_S,i,A_c,i of an arbitrary frequency component k_i of a parameter vector. It should be understood that this two- dimensional representation is chosen only because more than two dimen- sions are difficult to illustrate graphically; i.e. what the diagrams show are in fact two-dimensional projections of multi-dimensional vectors. The left-hand part of each diagram shows in solid lines components of a parameter vec- tor used by kinematic error calculator 13j to calculate the kinematic error compensation according to eq. (2); the right-hand part shows components of the parameter vector extracted from the torque data using eq. (1). Fig. 6A illustrates the case of step S3 where the parameter vector used by the kinematic error calculator 13j is 0, but a vector p #= 0 (of which compo- nents p_i are shown) is derived from the torque data in step S9.

Fig. 6B illustrates components

of the second parameter vector of the second feedback vector p resulting from

it. If there wasn’t the underlying imperfection that yields the first feedback vector p #= 0 for a parameter vector 0, the second parameter vector

could be expected to produce an ideal feedback vector

where

is related to

by an angle β_i between the two and a gain factor

are parameters of a transformation that would transform components p'_i of Fig. 6B into com- ponents

With these parameters known, it is possible to find a parameter vector p" which, by the same transformation, is transformed into -p (Fig. 6C).

This parameter vector p” is transmitted to kinematic error calculator 13j in step S13, so as to cause it, in the future, to calculate the kinematic error correction by inserting components of p" in eq. (2).

This process is carried out for one joint after the other. When afterwards (S14) the controller 2 enters production mode with all kinematic error calcu- lator 13_j, applying the correction of eq. 2, speed fluctuations of the reference point due to imperfections of the reduction gears 10j are substantially can- celled. Reference numerals

1 manipulator

2 controller

3 base

4 link

5 joint

6 axis

7 acceleration sensor

8 resolver

9 motor

10 harmonic drive gear

11 trajectory generator

12 joint controller

13 kinematic error calculator

14 adder

15 adder

16 wave generator

17 output shaft

18 flexspline

19 circular spline

Claims

1 . A method for reducing kinematic error in a joint (5_j) between a distal portion and a proximal portion of a manipulator (1), the joint (5_j) having associated to it a motor (9_j) mounted in one of said portions and coupled to the other one of said portions for driving rotation of the distal portion relative to the proximal por- tion by a reduction gear (10_j), the method comprising the steps of: a. providing a kinematic correction generator (13_j) which is pro- grammable using a parameter vector (0, p’, p") and output- ting, based on said parameter vector and a position (q_mot,j) input, a kinematic correction which is a sum of at

least one sinusoid having a frequency (k_i) which is a frequency of rotation of the motor (9_j) or of a periodic event in the reduction gear (10_j) occurring at a higher frequency than the rotation of the distal portion, or an integer multiple thereof, and an amplitude and a phase defined by said parameter vector (0, p’, p”), b. outputting to the motor a first drive control signal specifying a motor speed which is a sum of a standard speed

and a first kinematic correction

output by the kinematic correction generator (13_j) based on a first parameter vector (0); c. extracting, from an evaluation signal

received from the joint (5_j) in response to the first drive control signal, at least one frequency component having the frequency (k_i) of the at least one sinusoid, and determining a first feedback vector (p) defining a first feedback amplitude

and a first feed- back phase of said frequency component, d. outputting to the motor (9_j) a second drive control signal specifying a motor speed which is a sum of

the standard speed and a second kinematic correc

- tion output by the kinematic correction generator (13_j)

based on a second parameter vector (p’); e. extracting, from an evaluation signal

received from the joint (5_j) in response to the second drive control signal, a fre- quency component having the frequency (k_i ) of the at least one sinusoid, and determining a second feedback vector (p) defining a second feedback amplitude and a second

feedback phase of said frequency component,

f. subtracting said first feedback vector (p) from said second feedback vector

to obtain a feedback difference vector

g. finding a transformation which transforms said feedback dif- ference vector (p) into a difference (p’ - 0) between second and first parameter vectors, h. applying the transformation to the second feedback vector (p) to obtain a third parameter vector (p”), i. programming the kinematic correction generator (13j) using the third parameter vector (p”). The method of claim 1 , wherein the amplitude defined by said first parameter vector (0) is zero. The method of claim 1 or 2, wherein the standard speed

is a constant speed. The method of any of the preceding claims, wherein the step of extracting the frequency component having the frequency of the at least one sinusoid comprises high-pass filtering (S8) the eval- uation signal , a cutoff frequency optionally being lower than

the motor frequency and higher than the motor frequency multi- plied with the reduction ratio (R_j) of the reduction gear. The method of any of the preceding claims, wherein the stand- ard speed

or a harmonic thereof corresponds to a reso- nance frequency of the manipulator (1). The method of claim 5, further comprising at least one of the fol- lowing steps:

- controlling the distal portion to rotate at various speeds (S3) and choosing (S5) as the standard speed the one of

those various speeds where vibration of the manipulator (1) is strongest;

- controlling (S1) the distal portion to assume a configuration where its moment of inertia is maximized. The method of any of the preceding claims, wherein the evalua- tion signal is representative of motor torque.

The method of any of the preceding claims, wherein the reduc- tion gear (10_j) is a harmonic gear having a circular spline (19) with i_c teeth and a flexspline (18) with i_f teeth, and the fre- quency of the at least one sinusoid is times the rotation fre-

quency of the motor (9_j) or an integer multiple thereof. The method of claim 8, wherein the integer multiple is

times the rotation frequency. The method of any of the preceding claims, wherein the kine- matic correction is a sum of two or more sinusoids.

The method of any of the preceding claims, wherein the step of providing the kinematic correction generator (13j) comprises pro- gramming a processor to operate as the kinematic correction generator. A robotic system comprising

- a manipulator (1) having a proximal portion, a distal portion, a joint connecting (5_j) said proximal and distal portions and a motor (9_j) for driving rotation of the joint (5_j) , - a controller (12j) connected to the motor (9_j) and adapted to carry out at least steps b. to i. of the method according to one of claims 1 to 11. A computer software product comprising a plurality of machine- readable instructions which, when executed by a processor (13), cause the processor to carry out at least steps b.to i. of the method according to any of claims 1 to 11 .